OMEGA ALGEBRA MODEL OF CLINICAL WASTE INCINERATION

PROCESS

LIEW SIAW YEE

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy (Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

JUNE 2015

iii

To my beloved mother and father.

iv

ACKNOWLEDGEMENT

In particular, I would like to express my heartfelt appreciation to my

supervisor, Professor Dr. Tahir bin Ahmad, for his encouragement, guidance and

advice. Without his continued support and motivation, this thesis would not even

exist.

Next I wish to thank Ibnu Sina Institute for Fundamental Science Studies,

UTM for providing the facilities. Thank you to friendly administration stuffs who

always ready and willing to help.

Also thanks to my friends for their helping hand, advices and friendship

throughout the completion of this thesis.

Last but not least, I would love to thank my parents, my family members and

Andrew Jason Flavel for their love, understanding and continued emotional support.

v

ABSTRACT

This research shows how the Fuzzy Autocatalytic Set (FACS) is transformed

into a semigroup. Omega algebra is used as the main aspect to relate the FACS to the

transformation semigroup of omega algebra of the FACS. Clinical waste

incineration process is used as an example in supporting the developed theorems and

lemmas. C++ programming is used to develop a customized program that computes

the necessary data in supporting new theorems and lemmas. First, the FACS is

defined in terms of omega algebra, giving the omega algebra of FACS. Next, the

semigroup of omega algebra of FACS is constructed using the omega algebra of

FACS. The membership value of fuzzy edge connectivity is also determined to

complete the definition of the omega algebra of FACS. New definitions and

terminologies are used to model the clinical waste incineration process in terms of

omega algebra. As a result, the new model is shown to be able to comprehensively

explain the catalytic relation amongst the chemical elements in the clinical waste

incineration process. The established semigroup of omega algebra of FACS is then

defined to be a transformation semigroup of omega algebra of FACS. The results

show that FACS of the clinical waste incineration process and the structure of

transformation semigroup of omega algebra of the clinical waste incineration process

are “compatible”. In addition, the manifold representation of the transformation

semigroup of omega algebra of FACS is proposed for further studies.

vi

ABSTRAK

Kajian ini menunjukkan bagaimana Set Automangkinan Kabur (FACS)

ditransformasi kepada semikumpulan. Aljabar omega digunakan sebagai aspek

utama untuk menghubungkan FACS kepada transformasi semikumpulan aljabar

omega. Proses pembakaran sisa buangan klinikal digunakan sebagai satu contoh

untuk menyokong teorem dan lema yang telah dibangunkan. Pengaturcaraan C++

digunakan untuk membangunkan satu pengaturcaraan bagi mengira data yang

diperlukan dalam menyokong teorem dan lema baharu tersebut. Pada permulaanya,

FACS ditakrifkan dalam sebutan aljabar omega, yang memberikan aljabar omega

FACS. Seterusnya, semikumpulan aljabar omega FACS dibangunkan dengan

menggunakan aljabar omega FACS. Nilai keahlian kabur juga ditentukan untuk

menyempurnakan definisi aljabar omega FACS. Definisi dan istilah baharu ini

digunakan untuk memodelkan proses pembakaran sisa buangan klinikal. Natijahnya,

model baharu yang dibangunkan berupaya memaparkan hubungan pemangkinan

antara unsur kimia dalam proses pembakaran sisa buangan klinikal dengan lebih

menyeluruh. Semikumpulan aljabar omega FACS yang telah diperkenalkan

kemudiannya ditarifkan sebagai transformasi semikumpulan aljabar omega FACS.

Keputusan menunjukkan bahawa FACS dari proses pembakaran sisa buangan

klinikal dan struktur transformasi semikumpulan aljabar omega adalah “serasi”.

Sebagai tambahan, perwakilan manifold kepada transformasi semikumpulan aljabar

omega FACS telah dicadangkan untuk kajian pada masa hadapan.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF ABBREVIATIONS xiii

LIST OF SYMBOLS xiv

LIST OF APPENDICES xvii

1 INTRODUCTION 1

1.0 Background of the Research 1

1.1 Statement of the Problem 6

1.2 Objectives of the Study 6

1.3 Scope of Study 7

1.4 Significance of Study 7

1.5 Summary and Outline of Thesis 8

2 LITERATURE REVIEW 9

2.0 Introduction 9

2.1 Crisp Graph 9

2.2 Fuzzy Theory 14

2.2.1 Fuzzy Set and Fuzzy Number 14

viii

2.2.2 Fuzzy Relation and Fuzzy Interval Analysis 15

2.2.3 Fuzzy Graph 20

2.3 Autocatalytic Set and Fuzzy Autocatalytic Set 21

2.3.1 Autocatalytic Set 21

2.3.2 Fuzzy Autocatalytic Set 23

2.4 Fuzzy Autocatalytic Set of Clinical Waste Incineration

Process 25

2.5 Omega Algebra 26

2.6 Semigroup and Transformation Semigroup 26

2.7 Manifold, Tangent Space and Tangent Bundle 28

2.8 Conclusion 29

3 OMEGA ALGEBRA OF THE FUZZY

AUTOCATALYTIC SET 30

3.0 Introduction 30

3.1 Research Methodology 30

3.1.1 Omega Algebra and Graph 31

3.1.2 Programming Language C++ 32

3.2 Omega Algebra of FACS 33

3.3 Fuzzy Edge Connectivity of FACS 40

3.3.1 Membership Value of Fuzzy Edge Connectivity 42

3.4 Implementation: Omega Algebra Representation of

FACS 45

3.4.1 Omega algebra of FG 46

3.5 Conclusion 53

4 TRANSFORMATION SEMIGROUP OF OMEGA

ALGEBRA OF FUZZY AUTOCATALYTIC SET 55

4.0 Introduction 55

4.1 Transformation Semigroup of FACS 55

4.1.1 Transformation Semigroup of Omega Algebra of

Clinical Waste Incineration Process 58

4.2 Manifold Representation of ),( SQts FACS 62

ix

4.3 Conclusion 66

5 CONCLUSION AND FUTURE RESEARCH 68

5.0 Introduction 68

5.1 Conclusion of Thesis 68

5.2 Suggestions for Future Research 70

REFERENCES 71

Appendices A-E 75-107

x

LIST OF TABLE

TABLE NO. TITLE PAGE

2.1 Path between vertices of Graph 3G 13

3.1 Membership value of fuzzy edge connectivity between

any pair of vertices 45

3.2 Catalytic relation between vertices of FG 48

3.3 Paths and respective membership value of fuzzy edge

connectivity of 24* 51

3.4 Maximal membership value of fuzzy edge connectivity of FG 52

xi

LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Illustration of a clinical waste incinerator

(Sabariah et al., 2002) 2

1.2 A Graph showing the association between input-output

parameters and different components of the incineration process

(Sabariah et al, 2002) 3

1.3 A Crisp Graph, dG of the incineration process 4

1.4 Fuzzy Graph of Type 3 FdG for the clinical incineration process 5

2.1 Example of graph 10

2.2 Labeled and unlabeled graphs 10

2.3 Directed graph 3G 11

2.4 Examples of walks in 3G 11

2.5 Example of closed walk of 3G 12

2.6 Irreducible graph 13

2.7 Fuzzy graph ),( EVG 20

2.8 Image of H2O molecule 22

2.9 Image of Autocatalysis 22

2.10 Examples of Autocatalytic Set (ACS) 23

2.11 Fuzzy head and fuzzy tail 23

2.12 Tangent space on a point P in a manifold M 29

xii

3.1 -algebra for M 32

3.2 Research flow 34

3.3 Graphical representation of ( , )i jk v v 35

3.4 Unary operation for FACSFACS VV :1 35

3.5 Binary operation for FACSFACSFACS VVV :2 36

3.6 Ternary operation for FACSFACSFACSFACS VVVV :3 36

3.7 n-ary operation for 2n , FACS

n

FACSn VV : 37

3.8 Graphical representation of operation in FACS 38

3.9 Illustration of closure and associativity properties of in

the combustion chamber 40

3.10 Fuzzy head and tail of -operation 43

3.11 Example of membership value of fuzzy edge connectivity 44

4.1 Semigroup action and semigroup operation 57

4.2 Mapping between FACS of the clinical waste incineration

process (Sabariah, 2006) and transformation semigroup of

omega algebra of the clinical waste incineration process 59

4.3 Projection of for 22 )),(:(: ji vvW 63

4.4 Projection of for (a) 33 )),,(:(: kji vvvW and

(b) 44 )),,,(:(: lkji vvvvW 63

4.5 Projection of (a) 55 )),,,,(:(: mlkji vvvvvW and

(b) 66 )),,,,,(:(: nmlkji vvvvvvW 63

4.6 Projection of FGSW : 64

4.7 Path between elements ),( njmi vvvv 65

xiii

LIST OF ABBREVIATIONS

ACS - Autocatalytic Set

FACS - Fuzzy Autocatalytic Set

FGFACS - Fuzzy autocatalytic set of the clinical waste incineration

process

max - Maximal

max-av - Maximal average

max-prod - Maximal product

min - Minimum

PF(Q) - Partial function of Q

sup - Supreme

ts - Transformation semigroup

xiv

LIST OF SYMBOLS

rA~ - Assignment function of gradual number r~

nA~ - Assignment function of left profile over fuzzy interval N

nA~ - Assignment function of right profile over fuzzy interval N

],[ ba - Closed interval

],( ba / ),[ ba - Semi-open interval

C - Set of fuzzy edge connectivity

ijFC - Fuzzy edge connectivity between node i and node j

E - Set of edges

)( FwE - Edges have fuzzy weights

E(x, y) - Strength/ membership value of arc or edge between vertex x

and vertex y

ie - The thi edge/arc

G - Graph

dG - Crisp graph representation of clinical waste incineration

process

FG - Fuzzy graph representation of clinical waste incineration

process

i

FG - ith

type fuzziness of fuzzy graph

g - Mapping between fuzzy interval

xv

g - United extension of g

)( ieh - Fuzzy head of the thi edge

M - Manifold

N - Fuzzy interval

n~ - Fuzzy upper bound/ right profile over fuzzy interval N

n~ - Fuzzy lower bound/ left profile over fuzzy interval N

- Real number

r~ - Gradual number

TS, - Semigroup

),( S - Semigroup with semigroup operation

)(NS - Subset of fuzzy interval N

)( iet - Fuzzy tail of the thi edge

FGts - Transformation semigroup of clinical waste incineration

process

V - Set of vertices

V(x) - Strength / membership value of vertex x

FACSV - Set of vertices for FACS

iv - The thi vertex

kW - Walk with k-number of vertices

( , )i jk v v - Omega operation of k-operand from iv to jv

n - Omega operation of n-operand

X - Left endpoint of interval X

X - Right endpoint of interval X

+ - Addition function of vector space

xvi

• - Multiple function of vector space

-cut - Alpha-cut

- Composition of fuzzy relation

* - Omega operation of clinical waste incineration process

- Semigroup action

- Semigroup operation

- Element of

- Equal or greater than

- Every/ each

> - Greater than

- Less or equal

< - Less than

- Maximal

- Minimal

- Max-av composition

- Max-prod composition

)(xN - Membership function of fuzzy interval N

)( ij - Membership value of fuzzy edge connectivity between

vertex iv and vertexjv

)( ie - Membership value for fuzzy edge connectivity for edge i

-algebra - Set of omega algebra

FG - Omega algebra of clinical waste incineration process

FACS - Omega algebra of FACS

- Union

xvii

LIST OF APPENDICES

APPENDIX TITLE PAGE

A C++ Programming Code used to determine the

possible path of FG 75

B Possible path of every vertex of FG 82

C C++ Programming Code used to determine the

membership value of fuzzy edge connectivity of ( , )*i jk v v 88

D Membership value of fuzzy edge connectivity

of ( , )*i jk v v 98

E List of Publications 107

CHAPTER 1

INTRODUCTION

1.0 Background of the Research

From the time when Zadeh (1965) brought the inspiration of fuzzy set theory

by utilizing the concept of grade membership, many researchers and mathematicians

have been concerned with the properties and applications of fuzzy sets (Gitman &

Levine,1970; Tamura et al., 1971; Kandel and Yelowitz, 1974; Pathak and Pal, 1986;

Keller and Tahani, 1992). This is due to the fact that most crisp mathematical

models are always unable to model some details of reality including complexity and

ill-defined circumstances. Therefore the crisp models are generally insufficient in

describing the whole process of a system. On the other hand, the fuzziness concept

is proficient in modelling, explaining as well as predicting real life issues, such as

weather prediction (Riordan and Hansen, 2002). As a result, more and more studies

have been carried out by utilizing this idea.

The concept of fuzzy sets provide a natural way of dealing with problems in

which the source of imprecision is the absence of sharply defined criteria of class

membership rather than the presence of a random variable. The fuzzy graph was one

of the examples in fuzzy theory extensive application in its relation to graph theory.

Fuzzy graph theory was first introduced by Azriel Rosenfeld in 1975. Since then, it

has been quickly expanding and has several applications in various fields.

2

Modeling an incineration process is one of its applications to the real life

problem. Any waste or remaining supplies from medical, dental, veterinary,

pharmaceutical, treatment, hospital that may result in contamination to any

individual after exposure is defined as clinical waste. The waste incineration process

is a waste treatment process which underwent some controlled conditions which

included operating temperature, oxygen level, fuel supply and residence time.

Sabariah et al. (2002) embarked to model an incineration process of a

regional clinical waste incinerator facility in Malacca that is owned by Pantai

Medivest Sdn Bhd. Even though the incineration process seems simple, but as the

research was carried out in the year 2002, there was no essential mathematical model

to give details of the underlying principles involved in the process and operational

control of the incinerator. The schematic diagram of the facility is shown in Figure

1.1.

Figure 1.1 Illustration of a clinical waste incinerator (Sabariah et al., 2002)

A graphical representation in Figure 1.2 demonstrating the relation between

input-output variables of the incineration process was successfully achieved using a

3

graph. However, this graph model in Figure 1.2 was not able to describe the

incineration in detail.

Figure 1.2 A Graph showing the association between input-output parameter and

different components of the incineration process (Sabariah et al, 2002).

v1

v2

v3

v4

e1

e2

e3

e4

e5

e6

e7

e10 e11

e9

e12

e13

e14

e15

e19 e20

e8 e16

e18

e17

4

After some assumptions and refinements, a crisp model (Figure 1.3) was

created (Sabariah, 2006) whereby },,,,,{ 654321 vvvvvvV is the set of vertices

representing six variables that play a crucial part in the clinical waste incineration

process, namely waste, fuel, oxygen, carbon dioxide, carbon monoxide and other

gases including water, respectively.

Figure 1.3 A Crisp Graph, dG of the incineration process

The edges represent the connection between the variables in the process

which indicate their catalytic relationship and are denoted by the set E, ( , )i jv v which

means there is a catalytic relation between iv and jv where iv catalyzed the

production of jv . However, the crisp graph in Figure 1.3 was not able to illustrate

the waste incineration process in detail.

Due to some discrepancies of this crisp model (Figure 1.3) in explaining the

system has urged the study to look into the use of fuzzy theory, particularly the fuzzy

graph. Sabariah (2006) then applied the concept of the fuzzy graph into the

modeling of the clinical waste incineration process. The development and the results

of the model have been presented in Figure 1.4 (Tahir et al, 2010).

On the other hand, Blue et al. (1997; 2002) generalized a catalogue of various

possible types of fuzziness in a graph, which they named taxonomy of fuzzy graphs.

There are 5 types of graph in total that are characterized below:

5

Type 1: Fuzzy Set of Graphs,

Type 2: Crisp Set of Vertices and Fuzzy Edges,

Type 3: Crisp Set of Vertices and Edges with Fuzzy Connectivity,

Type 4: Fuzzy Vertex set and Crisp Edge Set,

Type 5: Crisp Graph with Fuzzy Weight.

The fuzzy graph in Figure 1.4 is proved to be a fuzzy graph Type 3 (Tahir et

al., 2010).

Figure 1.4 Fuzzy Graph of Type 3, FG for the clinical incineration process.

A clinical waste incineration process modeled using this new concept (Figure

1.4) was more precise than when modeled with a crisp graph (Figure 1.3) (Tahir et

al. 2010). As compared to the same color and thickness of each link in the crisp

graph (Figure 1.3), the different color of edges in Figure 1.4 signifies the different

range of membership values for the fuzzy edge connectivity. The changes in the

concentration of the chemical elements are demonstrated more precisely with the

changes in the thickness of the edges of each pair of elements in the process.

This graphical representation is found to be an Autocatalytic Set (ACS). A

new concept known as the Fuzzy Autocatalytic Set (FACS) was then fashioned

under the mishmash of fuzzy graph theory into ACS. Sabariah (2006) also showed

that this representation enlightens the clinical waste incineration process more

precisely.

6

The highlight of the investigation was the introduction of a fresh idea called

the Fuzzy Autocatalytic Set (FACS). This relation between fuzzy graphs to

autocatalytic sets has produced a few results in the forms of proven theorem

(Sabariah, 2006).

1.1 Statement of the problem

The crisp graph (Figure 1.3) representation of the clinical waste incineration

process gives the idea of connectivity between two elements of the process, while the

fuzzy graph representation of the clinical waste incineration process (Figure 1.4)

provides the changes of the concentration of the parameters during the process.

However, these two graphs representations were not able to tell the details of the

process as of yet. For example, the system cannot give information regarding the

catalytic interaction and relation amongst each and every parameter in the system,

not only among the pair.

In addition, the previous study was also unable to present all the possible

catalytic reactions between variables of the system. The main interest in this

research is the creation of the algebraic model to observe the system such that the

interaction between parameters can be observed and built more comprehensively.

1.2 Objectives of the Study

The goal of this research is to extend the algebraic structure of the fuzzy

autocatalytic set. In order to accomplish the goal, the following objectives are set as

research tactics. The objectives are:

a) to transform the graphical representation of the fuzzy autocatalytic set

to omega algebra.

7

b) to prove that the omega algebra of fuzzy autocatalytic set is a

semigroup.

c) to model the omega algebra of the fuzzy autocatalytic set in terms of

transformation semigroup.

d) to propose the manifold representation of the fuzzy autocatalytic set.

1.3 Scope of the Study

For this research, the emphasis is on the exploration of algebraic structures

behind the clinical incineration reaction process that was modeled using FACS. The

main interest is to model this model of FACS into few possible algebraic structures

in order to study and algebraically explain the changes of the variable.

The omega algebra concept would be the foundation and the main aspect of

the proposed algebraic model. The algebraic model assembled will then be analyzed

in terms of validity in explaining the clinical waste incineration process.

1.4 Significance of Study

The main aim of this research is to extend the algebraic structure of the fuzzy

autocatalytic set of the fuzzy graph. The omega algebra concept would be the

foundation and the main aspect of the proposed algebraic model of the clinical waste

incineration process. This new structure is called omega algebra of fuzzy

autocatalytic set of the clinical waste incineration process. The new structure

subsequently yields some physical interpretations in the form of proven lemmas and

theorems for the clinical incineration process which has been modeled as a fuzzy

autocatalytic set.

8

1.5 Summary and Outline of Thesis

The first chapter provides the general background and information regarding

the research.

Chapter 2 gives the published literature of relevant topics to the research

which includes graph theory, fuzzy theory, fuzzy graph, the autocatalytic set and

Fuzzy Autocatalytic Set, the Fuzzy Autocatalytic Set of the clinical waste

incineration process as well as omega algebra, the transformation semigroup and

manifold.

The third chapter presents the construction of the omega algebra structure of

the Fuzzy Autocatalytic Set. Then the omega algebra of the Fuzzy Autocatalytic Set

is applied into clinical waste incineration as an example in explaining this algebraic

structure. The same chapter also provides the definition of the membership value of

fuzzy edge connectivity of the omega algebra of the Fuzzy Autocatalytic Set of the

clinical waste incineration process.

Chapter 4 presents the extension algebraic model of omega algebra of the

Fuzzy Autocatalytic Set which is constructed in the previous chapter. This includes

the transformation semigroup of omega algebra of the Fuzzy Autocatalytic Set and

manifold representation of transformation semigroup of omega algebra of the clinical

waste incineration process.

Chapter 5 concludes the results and findings presented in the preceding

chapters and suggest several ideas for further research on this topic.

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2 2(T , , )S S